Preprint 2009-052

Helge Holden and Xavier Raynaud

Abstract:
We prove the existence of a global semigroup
for conservative solutions
of the nonlinear variational wave equation
utt−c(u)(c(u)ux)x=0.
We allow for initial data
u|t=0 and
ut|t=0
that contain measures.
We assume that
0<κ−1≤c(u)≤κ.
Solutions of this equation may experience
concentration of the energy density
(ut2+c(u)2ux2)dx
into sets of measure zero.
The solution is constructed
by introducing new variables related to the characteristics,
whereby singularities in the energy density become manageable.
Furthermore, we prove that
the energy may only focus on a set of times of zero measure
or at points where c'(u) vanishes.
A new numerical method to construct conservative solutions
is provided and illustrated on examples.

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